December 12, 2002 UCB-PTH-02/59 SLAC-PUB-9612 hep-ph/0212180 Atmospheric Neutrinos Can Make Beauty Strange∗ Roni Harnik, Daniel T. Larson and Hitoshi Murayama Theoretical Physics Group Ernest Orlando Lawrence Berkeley National Laboratory University of California, Berkeley, California 94720, USA Department of Physics, University of California Berkeley, California 94720, USA Aaron Pierce Theoretical Physics Group Stanford Linear Accelerator Center Stanford University, Stanford, California,94309, USA Abstract The large observed mixing angle in atmospheric neutrinos, coupled with Grand Unification, motivates the search for a large mixing be- tween right-handed strange and bottom quarks. Such mixing does not appear in the standard CKM phenomenology, but may induce signifi- cant b → s transitions through gluino diagrams. Working in the mass eigenbasis, we show quantitatively that an O(1) effect on CP violation in B0d → φKS is possible due to a large mixing between s̃R and b̃R, while still satisfying constraints from b → sγ. We also include the effect of b̃L-b̃R mixing proportional to mbµ tan β. In the case where mbµ tan β � M 2SUSY there may be a large effect in Bs mixing corre- lated with a large effect in B0d → φKS, typically ∆MBs >∼ 100 ps−1, an unambiguous signal of new physics at Tevatron Run II. ∗The work of RH, DL, and HM was supported in part by the U.S. Department of Energy under Contract DE-AC03-76SF00098, and in part by the National Science Foundation under grant PHY-0098840. The work of AP was supported by the U.S. Department of Energy under Contract DE-AC03-76SF00515 1 Introduction Flavor physics has seen tremendous progress in the past few years. The discovery of neutrino oscillations by the SuperKamiokande [1], SNO [2], and KamLAND [3] experiments clearly marks a historic event, while CP violation has recently been found in two new manifestations: direct CP violation in the neutral kaon system [4] and indirect CP violation in the the B0d system [5]. On the other hand, we still lack insight into the origin of flavor and the patterns of masses and mixings. We need to look for any possible hints of physics that give us additional insight into these questions. One of the major surprises in neutrino physics was the observation of (two) large angles. Unlike in the quark sector where all mixing angles in the Cabibbo–Kobayashi–Maskawa (CKM) matrix are small, both atmospheric and solar neutrino oscillations require large angles. An important question is whether the presence of large angles will give us new insight into the origin of flavor, masses, and mixings. It was pointed out in Ref. [6] that the large angles in the neutrino sector may imply large angles in the mixing among right-handed down-type quarks if they are grand-unified with lepton doublets. Indeed, some SO(10) models with Pati–Salam type unification of Yukawa matrices suggest that the large mixing angles in neutrinos arise from the charged lepton mass matrices, and thus also appear in the down-quark mass matrices. In these models, one assumes that these new large mixing angles do not appear in the CKM ma- trix because the right-handed charged-current interaction is broken at the Pati-Salam unification scale. However, the imprint of the large atmospheric neutrino mixing angle may appear in the squark mass matrices as a large b̃R-s̃R mixing effect though radiative corrections due to the large top Yukawa coupling. The large solar neutrino mixing angle, however, does not cause a significant effect because of the smaller Yukawa coupling for lower genera- tions. The new b̃R-s̃R mixing in turn feeds into new effects in B-physics. In particular, there may be large new CP-violating effects in b → s transitions and enhanced Bs mixing. It has already been noted that CP violation in B0d → φKS is a good place to look for new physics effects [7, 8]. The time-dependent asymmetry in B0d(B 0 d) → φKS was reported recently by both BaBar and BELLE. Their measurements differ from the value in the J/ψKS final state by O(1). The standard model predicts that these two channels should give the same value. The significance of the difference is 2.7σ if the measurements from both collaborations are combined– the current 1 world average for sin 2β in the B0d → J/ψKS channel is 0.734 ± 0.054, while in the B0d → φKS channel SφK = −0.39 ± 0.41 [9]. This report has already sparked many speculations [10]. It is not clear if this is a temporary anomaly or a genuine new effect. Nonetheless it is important to study how large the new CP violation in B0d → φKS can be and how it is correlated to Bs mixing which will be studied soon at Tevatron Run II. In this paper, we investigate the size of CP violation in B0d → φKS as well as Bs mixing from a potentially large b̃R-s̃R mixing. There have also been several investigations of B0d → φKS within the context of supersymmetry (SUSY) [8, 11, 12]. Of the above, only Ref. [12] investigated the correlation between the measurement of sin 2β in B0d → φKS and Bs mixing. However it uses the mass insertion formalism, which is not necessarily appropriate for the large mixing that we will consider. In addition, it appeared before the recent experimental results, and so it did not seek to reproduce such a large shift in sin 2β. We perform a calculation in the mass eigenbasis, with a goal of determining whether supersymmetry can accommodate the central value of the recent experimental results for sin 2β in B0d → φKS. We then explore the consequences for Bs mixing. We also emphasize contributions to sin 2β that arise from a combination of b̃L-b̃R (mbµtanβ) and b̃R-s̃R mixing. These contributions, which we find to be important over a wide region of parameter space, are not easily analyzed in the mass insertion approximation. Analogous combinations were studied in the kaon system [13], but to our knowledge these contributions have not been thoroughly analyzed with regard to new physics in the B0d system. The outline of the paper is as follows. In the next section we introduce the effective field theory formalism for b decay and work out b → s transitions. In Section 3 we discuss Bs mixing from large b̃R-s̃R mixing. Section 4 is devoted to the discussion of correlations between the b → s transition and Bs mixing. We conclude in Section 5. Details of some calculations are presented in the appendices. In Appendix A we show the loop functions, while the hadronic matrix elements are estimated in Appendix B. 2 CP Violation in b → s Transition In this section we briefly review the well-known effective field theory formal- ism for B-physics (for a comprehensive review see [14]), which we use to calculate the contribution of supersymmetric particles to b → s transitions. 2 Using this machinery we discuss the contribution of a large mixing between right-handed squarks to the CP-violating parameter SφK. We also use this formalism to address the constraints on the SUSY contribution that come from the b → sγ radiative decay. 2.1 Effective Hamiltonian The b → s transitions of interest can be described by the following ∆B = 1 effective Hamiltonian: Heff = 6∑ i=1 (CiOi + C′iO′i) + CγOγ + C′γO′γ + CgOg + C′gO′g, (1) where O1 = (s̄iγµPLcj)(c̄jγµPLbi) (2) O2 = (s̄γµPLc)(c̄γµPLb) (3) O3 = (s̄γµPLb)(s̄γµPLs) (4) O4 = (s̄iγµPLbj)(s̄jγµPLsi) (5) O5 = (s̄γµPLb)(s̄γµPRs) (6) O6 = (s̄iγµPLbj)(s̄jγµPRsi) (7) Oγ = e 8π2 mb(s̄iσ µνPRbi)Fµν (8) Og = gs 8π2 mb(s̄iσ µνTaijPRbj)G a µν. (9) Here i and j are color indices (suppressed in color singlet terms), PR,L = (1± γ5)/2, and σ µν = i 2 [γµ,γν]. The primed operators, which are not generated at leading order in the Standard Model, are obtained by taking L ↔ R everywhere. Here we have ignored the electroweak penguin operators O7−10 and the contributions to the dipole operators proportional to the s-quark mass, ms. Following the standard procedure for incorporating QCD corrections we match the Wilson coefficients at a high scale to loop diagrams containing heavy particles present in the full theory, and then use the renormalization group equations (RGE) to run the coefficients to the low scale where mesons decay. We incorporate leading order QCD corrections using the anomalous dimension matrices given in [14]. The initial conditions for the standard 3 bR g ~ g ~ sR bR ~ s s s s R,L R,L R,L R ~ ~ δ d 23( )RR _ b b R R ~ sR ~ s s s s R,L R,L R R ~ (δ23)RR d _ b sRR ~~ b b L R sR δ d 23( )RR mb g b b L ~ sR R ~ ~ bL sR δ d 23( )RR tanβµmb m~g g Figure 1: Box and penguin contributions to the b → ss̄s transition. The bottom row shows contributions to the chromo-dipole operator. We show the mass insertions for pedagogical purposes but perform calculations in the mass eigenbasis. model coefficients are also in [14]; only C2, Cγ and Cg are nonzero at leading order. Leading order running of the standard model coefficients has mixing between all eight operators, O1−6,γ,g, due to the presence of a tree level contribution to O2. Since right-handed squark mixing only contributes to the primed operators and gives no tree level contributions, the leading order SUSY running is simpler: O′3−6 mix only amongst themselves, as do O′γ,g. The SUSY contributions come from box, penguin, and dipole diagrams. Figure 1 shows sample diagrams with mass insertions schematically indicat- ing the mixing. However, since we are allowing for large mixing between the 2nd and 3rd generation squarks, we use the mass eigenbasis for our compu- tations. Furthermore, we find that the region where the effect on SφK in the B0d → φKS channel is maximized is a region where the squarks are highly non-degenerate, again calling into question the validity of the mass insertion approximation.1 The squark mass matrix we consider is motivated by models where a large right-right mixing between the second and third generations is expected, such 1 Indeed, our results differ somewhat from previous investigations done in the mass insertion approximation, e.g. [12]. 4 as in [6] where this mixing is related to the large mixing in atmospheric neutrinos. In addition to this new contribution we must include other off- diagonal terms that already exist in the Minimal Supersymmetric Standard Model (MSSM), namely q̃Rq̃L couplings induced by the cross term between the Yukawa couplings and the µ term.2 Of the down type squarks only the third generation can have appreciable left-right mixing, which is proportional to mbµtanβ. Thus the mass matrix takes the approximate form M̃2 d̃ =   m̃2Ld 0 0 0 0 0 0 m̃2Ls 0 0 0 0 0 0 m̃2Lb 0 0 mbµtanβ 0 0 0 m̃2Rd 0 0 0 0 0 0 m̃2Rs m̃ 2 Rsb 0 0 mbµ ∗ tanβ 0 m̃∗2Rsb m̃ 2 Rb   . (10) We define the mass eigenvalues and mixing matrices as follows: U†M̃2 d̃ U = diag(m̃2L1,m̃ 2 L2,m̃ 2 L3,m̃ 2 R1,m̃ 2 R2,m̃ 2 R3), (11) where U is a unitary rotation matrix. Without loss of generality we assume that m̃R3 ≤ m̃R2 (we allow an arbitrary mixing angle). In the mass eigenbasis 3×6 mixing matrices, ΓL,R, appear in the quark-squark-gluino vertices. They are related to U by UaA = ( ΓL∗iA ΓR∗iA ) , (12) where i = d,s,b labels the gauge eigenstates, A = 1, . . . ,6 labels the mass eigenstates, and the index a labels states in the basis (dL sL bL dR sR bR). To investigate the effect of 2nd and 3rd generation mixing of the right-handed squarks we parameterize the mixing matrix as follows: U = Φ(φ5, φ6)R36(θ36)R35(θ35)R56(θ56), (13) where Φ = diag(1,1,1,1,eiφ5,eiφ6) is a phase matrix, and Rij(θij) is a 2 × 2 rotation in the ij plane. The angle θ35 can be solved for using our assumption that there is no mixing between b̃L and s̃R. The Wilson coefficients in the 2For simplicity we ignore terms that may arise from trilinear soft terms. 5 mass eigenbasis were previously given in [11, 15]. We reproduce them here, correcting two typographical errors. C′3 = α2s m2g̃ (∑ AB ΓR∗sAΓ R bAΓ R∗ sBΓ R sB [ −1 9 B1(xA,xB) − 5 9 B2(xA,xB) ] + ∑ A ΓR∗sAΓ R bA [ − 1 18 C1(xA) + 1 2 C2(xA) ]) (14) C′4 = α2s m2g̃ (∑ AB ΓR∗sAΓ R bAΓ R∗ sBΓ R sB [ −7 3 B1(xA,xB) + 1 3 B2(xA,xB) ] + ∑ A ΓR∗sAΓ R bA [ 1 6 C1(xA) − 3 2 C2(xA) ]) (15) C′5 = α2s m2g̃ (∑ AB ΓR∗sAΓ R bAΓ L∗ sBΓ L sB [ 10 9 B1(xA,xB) + 1 18 B2(xA,xB) ] + ∑ A ΓR∗sAΓ R bA [ − 1 18 C1(xA) + 1 2 C2(xA) ]) (16) C′6 = α2s m2g̃ (∑ AB ΓR∗sAΓ R bAΓ L∗ sBΓ L sB [ −2 3 B1(xA,xB) + 7 6 B2(xA,xB) ] + ∑ A ΓR∗sAΓ R bA [ 1 6 C1(xA) − 3 2 C2(xA) ]) (17) C′γ = αsπ m2g̃ (∑ A ΓR∗sAΓ R bA [ −4 9 D1(xA) ] + mg̃ mb ∑ A ΓR∗sAΓ L bA [ −4 9 D2(xA) ]) (18) C′g = αsπ m2g̃ (∑ A ΓR∗sAΓ R bA [ −1 6 D1(xA) + 3 2 D3(xA) ] + mg̃ mb ∑ A ΓR∗sAΓ L bA [ −1 6 D2(xA) + 3 2 D4(xA) ]) . (19) Here we use the definition xA = m̃ 2 A/m 2 g̃, where mg̃ is the gluino mass. The loop functions are given in Appendix A. Note that the contributions due to left-right mixing only enter the dipole operators C′γ and C ′ g where they are enhanced by a factor of mg̃/mb over the right-right mixing contributions to the same operators. Also notice that with our choice of the mixing matrix, U, the mass eigenvalues m̃L1 and m̃R1 do not enter the Wilson coefficients. Because there was some disagreement in the literature, we have explicitly recomputed the box contributions (proportional to B1 and B2). However, the 6 penguin contributions (proportional to C1,2 and D1−4) are well established. See for example, [16]. We found several inconsistencies in the literature which can be remedied as follows. In Equation (A.8) of [11] the coefficient − 1 4Nc of B1 should be replaced by − 1Nc and the expression for C̃ DM R in Equation (A.16) should be multiplied by − i 2 . In Equation (42) of [12] the factor of 7 18 should be 7 6 , and in Equation (43) the factor of 16 9 should be 8 9 . The loop function M2(x) in [16] should be multiplied by −x instead of − 1x. Finally, in [8] there are typos in each line of Equations (B.4a-e) and in (B.6a,b). 2.2 B0d → φKS We now specialize our discussion to the B0d → φKS decay, with the goal of computing the contribution to the CP asymmetry measured in this chan- nel. In addition to the Wilson coefficients we have presented, we must also compute the hadronic matrix elements of the operators. The calculation of these matrix elements is non-perturbative, so approximations must be made in order for us to make progress. In the naive factorization approximation we break each matrix element up into a pair of color singlet currents, one which creates the φ from the vacuum and the other that mediates the B0d → K decay, and we discard any color-octet currents. For the operators O(′)3−6 there are two ways of contracting the external quarks with the quark fields in the operator. After employing Fierz trans- formations as necessary, and using the identity δijδk� = 1 N δi�δkj + 2T a i�T a kj to rearrange color indices to form singlet currents, we arrive at the following matrix elements [8]: 〈φKS|O3,4|B̄0d〉 = 1 4 H ( 1 + 1 Nc ) (20) 〈φKS|O5|B̄0d〉 = 1 4 H (21) 〈φKS|O6|B̄0d〉 = 1 4 H 1 Nc (22) where H = 2(,φ·pB)fφm2φF+(m2φ) (see Appendix B for definitions of the decay constant and form factors). The same results hold for the matrix elements of the corresponding primed operators because the axial vector currents do not contribute, so the chirality of the operators is irrelevant. We take Nc = 3 throughout our analysis. 7 The matrix elements of the chromo-dipole operators O(′)g are more difficult to analyze, so we show the details explicitly in Appendix B following [8]. These manipulations yield 〈φKS|Og4|B̄0d〉 = κ αs 2π H N2c − 1 2N2c . (23) where our definition of κ agrees with [11] up to a sign convention.3 Nu- merically we find κ = −9 8 + O ( m2 φ m2 B ) � −1.1. However, there were many assumptions about the quark momenta that go into this estimation of κ, so the numerical value of −1.1 should be taken as a guideline only. We will present our results for various values of κ to demonstrate the dependence. There is one final ingredient in the Standard Model contribution to the amplitude. This comes from the one-loop matrix element of O2 when the charm quarks are closed into a loop. It is given by [17] P = αs 8π C2( 10 9 + G(mc,µ,q 2)) with G(m,µ,q2) = 4 ∫ 1 0 dxx(1 − x) ln m 2 − x(1 − x)q2 µ2 . (24) Numerically we use mc = 1.35 GeV and q 2 = m2 b 2 which gives P = (−0.015 − 0.011i)C2. Putting the factorized matrix elements together the amplitude for B̄d → φKS can be written ĀSMφK = H [ 1 4 ( 1 + 1 Nc ) (C3 + C4) + 1 4 C5 + 1 4 1 Nc C6 ] + H [ 8 9 P + κ αs 2π N2c − 1 2N2c Cg ] (25) ĀSUSYφK = H [ 1 4 ( 1 + 1 Nc ) (C′3 + C ′ 4) + 1 4 C′5 + 1 4 1 Nc C′6 ] + H [ κ αs 2π N2c − 1 2N2c C′g ] (26) 3Note that in [11] the overall factor of 1 4 in the first line of Equation (18) should not be multiplying the last term that contains κDM. 8 The time-dependent CP-asymmetry is given by aφK(t) = CφK cos(∆MB0 d t) + SφK sin(∆MB0 d t), (27) where CφK = 1 − |λ|2 1 + |λ|2 , SφK = 2 Imλ 1 + |λ|2 . (28) Here λ is defined as λ = q p A(B̄0d → φKS) A(B0d → φKS) = q p (ĀSMφK + ĀSUSYφK ) (ASMφK + ASUSYφK ) . (29) The ratio q/p from B0d mixing is dominated by the standard model and is nearly a pure phase, ei2β, where β is the standard angle of the unitarity triangle. In the Standard Model the ratio of amplitudes Ā/A is real, i.e. there is no CP violation in the decay, rather all CP violation results from mixing. On the other hand, O(1) phases in the supersymmetric contribution can give the ratio a phase, φSUSY. Then we have λ = ei(2β+φ SUSY) |Ā| |A| ⇒ SφK = sin(2β + φ SUSY). (30) Thus the presence of a phase in the down squark mixing matrix can alter the measured value of SφK from the standard model prediction of SφK = sin 2βJ/ψK = 0.73. The amount of deviation is described in Section (2.4). Also note that P possesses a strong phase that is not present in ASUSY. The presence of a weak phase in ASUSY then allows for the possibility of nonzero direct CP-violation, namely CφK �= 0. We do not pursue this sig- nature further here, as quantitative statements are difficult due to the large hadronic uncertainties. 2.3 Constraints from b → sγ A large mixing between right-handed strange and bottom squarks generates the operator O′γ through penguin diagrams, as in Figure 2. Therefore the tight experimental constraints on the branching ratio BR(B → Xsγ) serve to limit the contributions from squark mixing. In the model we consider there are two important contributions to the O′γ operator. We can classify the contributions according to where the helicity 9 b s R R~ ~ b b R L sR γδd23( )RR mb b b L ~ sR R ~ ~ bL sR δ d 23( )RR γ m~g tanβµmb Figure 2: Sample contributions to the b → sγ transition. We show the mass insertions here for pedagogical purposes, but perform calculations in the mass eigenbasis. flip for O′γ arises. In the first contribution, the helicity flip is present on the external b-quark line, and gives a contribution proportional to the b-mass. This contribution is present even when the only mixing between the squarks is an off-diagonal mixing between the right-handed squarks of the second and third generation. The constraint on this contribution is relatively mild.4 The second contribution has a helicity flip on the gluino line, so is enhanced relative to the first contribution by a factor of mg̃/mb (see Equations (18) and (19)). This contribution is only present if there is left-right mixing in the squark matrix. Because of the mg̃/mb enhancement, this contribution is relatively strongly constrained. In our framework, this contribution arises only from the combination of a left-right mixing between the b̃ squarks and the right-right mixing between the b̃ and s̃ squarks. The result is that for large values of µtanβ, a smaller b̃R-s̃R mixing is allowed. When there is no significant off-diagonal mixing among the left-handed squarks, we can write: BR(b → sγ) ∝ |Cγ|2 + |C′γ|2, where the first contri- bution is from the standard model and the second is from supersymmetric penguins.5 A recent theoretical evaluation within the Standard Model gives [18]: BR(b → sγ)th = (3.60 ± .30) × 10−4. (31) After rescaling to limit the photon energies to Eγ > 1.6 GeV (for details see [18]), an averaging of experimental results from BaBar, BELLE, CLEO, and 4We note that the limits of reference [16] assume that new contributions to b → sγ are summed incoherently. In general, this will underestimate the contribution of the LL mass insertion. 5The two contributions are added incoherently because they contribute to different final helicity states of the s-quark. 10 ALEPH [19] yields BR(b → sγ)exp = (3.29 ± .34) × 10−4. (32) The experimentally measured branching ratio is actually slightly smaller than the standard model prediction, which leaves little room for new physics con- tributions. Subtracting experiment from theory we find: BR(b → sγ)th −BR(b → sγ)exp = (.31 ± .45) × 10−4. (33) We will require that the supersymmetric contribution keep the theoretical prediction within 2σ of the experimentally measured value. This means ad- ditional contributions from supersymmetry can be roughly 1/6 of those in the Standard Model. For simplicity, and to avoid the theoretical uncertainty associated with the direct calculation of the branching ratio, we will con- strain the supersymmetric contributions by requiring |C′γ|2 ≤ 0.16 × |Cγ|2 where both coefficients are calculated to leading order. Thus we are making the simplifying assumption that the higher order QCD corrections affect the two operators in the same way. 2.4 Numerical Analysis Within the framework we have chosen, motivated by atmospheric neutrino oscillations, there are four mass eigenvalues, two mixing angles, and two phases in the down-squark mass matrix that enter in the computations of SφK and BR(b → sγ). However, the fact that the neutron electric dipole moment (EDM) has not been observed strongly constrains the phase of µ, especially for large µtanβ. We have checked that allowing a non-zero phase of µdoes not substantially affect even our quantitative conclusions. Therefore we conservatively take the phase of µ to be zero for the remainder of this paper. Including the gluino mass we are then left with eight essentially unknown parameters. In order to reduce the size of the parameter space we will investigate two limiting cases: the case where the (flavor-diagonal) mixing between left- and right-handed b̃ squarks is negligible and the case where such mixing, when coupled with the large right-right mixing, leads to the dominant contribution. We refer to the latter case as “LR+RR mixing.” Note that by taking µ to be real, the remaining phase only appears as an overall phase in ΓR. As a result, all SUSY diagrams have the same phase. 11 2.4.1 Dominant Right-Right Mixing First we consider the situation where the contribution from b̃L-b̃R mixing is negligible, i.e. mbµtanβ � m̃2, the mass scale of the squarks.6 Here the parameter space is reduced: in this limit there is only one mixing angle, one phase, and three mass eigenvalues that enter the computation of the Wilson coefficients. The presence of mixing with an order one phase in the right-handed down squark sector can significantly alter the measured value of SφK. Our first question is whether a large right-right mixing between the down squarks can reproduce the central value for sin 2β in the φKS channel measured at the B-factories. We find that using the central value of our estimate for κ = −1.1, it is difficult to reproduce the observed central value and accommodate the constraints from b → sγ. However, this estimate for κ is highly uncertain, and increasing the magnitude of κ increases the contribution to SφK without changing the contribution to b → sγ. Therefore, we present our numerical results for two cases, κ = −1.1, and a value with greater magnitude, κ = −2, which we still view as reasonable given the substantial uncertainties involved in its estimation. In Figure 3 we show contours of SφK as a function of the gluino mass and m̃R3. We have also chosen values for the mixing angle and phase in Γ R which give the greatest deviation of SφK from the Standard Model prediction. Also shown are contours of the percent increase in BR(b → sγ) due to new physics and the corresponding values of ∆MBS (the latter will be discussed in Section 3). Generally speaking, lighter squark and gluino masses increase the effect of the new physics contributions, allowing SφK to depart from the standard model expectation. But at the same time this increases the contribution to BR(b → sγ) and runs up against the experimental constraint. In Figure 4 we plot the same contours as a function of the gluino mass and the heavier squark mass m̃R2, with m̃R3 = 300 GeV. For κ = −2.0 there is a range of gluino masses where SφK can be below zero while the increase in BR(b → sγ) is less than 16%. The minimum possible SφK allowed by this constraint decreases as the magnitude of κ increases. Finally in Figure 5 we plot the same contours as a function of the gluino 6Within the MSSM µ tanβ cannot go to zero while satisfying experimental constraints. We find, however, that there is a portion of parameter space above the smallest exper- imentally allowed value of µ tanβ where the right-right mixing diagrams are dominant. Furthermore, right-right mixing dominates when m̃R � m̃L. 12 150 200 250 300 350 400 450 500 150 200 250 300 350 400 450 500 −0.2 0.2 200 ps 300 ps +10% +16% m~ m g~ (GeV) (GeV) −1 −1 400 ps−1 0.5 0.0 −0.4 +22% 0.4 150 200 250 300 350 400 450 500 150 200 250 300 350 400 450 500 200 ps 300 ps 400 ps −0.2 −0.4 0.5 0.4 0.2 +10% 0.0 (GeV) m g~ ~m (GeV) −0.6 −1 −1 −1 +22% +16% Figure 3: Contours of SφK (thin blue), percent increase in BR(b → sγ) (thick green), and ∆MBs (dashed red) for m̃R2 = 5 TeV, optimal mixing angle and phase, and κ = −1.1 (left) and κ = −2.0 (right). mass and κ with m̃R2 = 5 TeV and m̃R3 = 300 GeV. The b → sγ constraint is independent of κ, and we see how SφK decreases with the increasing mag- nitude of κ. 2.4.2 Dominant LR + RR Mixing Now we consider a second limiting case, where the diagonal left-right mixing leads to the dominant contribution to the b-quark decays. The contribution fromLR+RR mixing is enhanced in the dipole operators, so we may focus our attention on the coefficients C′g and C ′ γ. To evade the constraint from b → sγ, while simultaneously getting a large effect in SφK, we want to minimize the ratio C′γ/C ′ g, which can be done by taking larger values of x, i.e. squark masses much heavier than the gluino. In Figure 6 we reproduce Figure 3 but with a large value of µtanβ ∼ 35 TeV. In this region the lightest squark mass eigenvalue, m̃R3, needs to be above 1 TeV to avoid the bound from b → sγ. For κ = −1.1 the smallest SφK can be is about 0.3 for a small gluino mass, though this can get as low as −0.2 for κ = −2.0. In this case the κ-dependence is very simple because the main contribution to SφK comes from a single operator O′g whose contribution is directly proportional to κ. Thus an increased absolute value of κ directly increases the effect in SφK without affecting the bound from b → sγ. 13 0 1000 2000 3000 4000 5000 6000 150 200 250 300 350 400 450 500 m 300 ps ~ −1 −1 +22% −0.2 0.0 −0.4 0.2 0.4 0.5 +16% +10% 400 ps mg~ (GeV) (GeV) R2 100 ps 200 ps −1 −1 0 1000 2000 3000 4000 5000 6000 150 200 250 300 350 400 450 500 −0.4 −0.2 0.2 0.5 m −1 −1 −1 +22% +16% +10% 0.4 0.0 (GeV) m g~ −1100 ps 200 ps 300 ps 400 ps (GeV)R2 ~ Figure 4: Contours of SφK (thin blue), percent increase in BR(b → sγ) (thick green), and ∆MBs (dashed red) for m̃R3 = 300 GeV, optimal mixing angle and phase, and κ = −1.1 (left) and κ = −2.0 (right). −3 −2.5 −2 −1.5 −1 −0.5 0 150 200 250 300 350 400 450 500 0.5 0.4 0.0 κ −1 0.2 −0.2 −0.4 350 ps−1 250 ps 450 ps−1 m ~g (GeV) +22% +16% +10% Figure 5: Contours of SφK (thin blue), percent increase in BR(b → sγ) (thick green), and ∆MBs (dashed red) for m̃R2 = 5 TeV, m̃R3 = 300 GeV, and optimal mixing angle and phase. 14 1000 1200 1400 1600 1800 200 400 600 800 1000 gm ~ (GeV) m~ R3 (GeV) 40 ps−1 15 ps −1 20 ps−1 +8%+16% +40% +24% −0.4 0.0 0.6 −0.2 0.2 0.4 1000 1200 1400 1600 1800 200 400 600 800 1000 gm ~ (GeV) m~ (GeV) 40 ps−1 20 ps−1 15 ps−1 +16% +8%+24% +40% 0.0 0.2 0.4−0.6 −0.4 −0.2 Figure 6: Contours of SφK (thin blue), percent increase in BR(b → sγ) (thick green), and ∆MBs (dashed red) for m̃L2, m̃L3, m̃R2 ∼ 2 TeV, optimal mixing angles and phases, µtanβ ∼ 35 TeV, and κ = −1.1 (left) and κ = −2.0 (right). 2.4.3 Combination of Contributions In order to ascertain the relevance of each of these two regimes we scanned over the parameter space searching for the minimal values of SφK as a func- tion of the product µtanβ. The result is shown in Figure 7 for κ = −1.1 and κ = −2.0. From the figure it is clear that for κ = −1.1 there is a slightly larger effect on SφK for smaller values of µtanβ, though the entire region allows for a substantial deviation from the standard model. As expected, increasing the magnitude of κ increases the effect on SφK somewhat more for the larger values of µtanβ where the chromo-dipole operator gives the main contribution. To get a sense for the relative size of the RR and LR+RR contributions to SφK, we can compare the magnitude of the two terms comprising C ′ g, Equation (19). Not surprisingly, for µtanβ greater than about 25 TeV the LR + RR contribution dominates by an order of magnitude. However, even for µtanβ < 1 TeV there can be points where the LR + RR contribution to the chromo-dipole operator is just as important as the RR contribution. This underscores the importance of treating this calculation in the mass eigenbasis. 15 10 20 30 40 50 60 −0.4 −0.2 0.2 0.4 0.6 m in (S ) tan βµ (TeV) κ=−1.1 κ=−2.0 φ K Figure 7: Minimum values of SφK as a function of µtanβ resulting from a scan of 58 million points that satisfied the b → sγ constraint. The upper set of points has κ = −1.1 while the lower set has κ = −2.0. 3 Bs Mixing Mixing between b̃R and s̃R also leads to a significant contribution to Bs-B̄s mixing. In our scenario the effective Hamiltonian that receives such contri- butions consists of three operators that have nonzero coefficients: HSUSYeff = CVRR(s̄γµPRb)(s̄γµPRb) + CSLL(s̄PLb)(s̄PLb) + CS×LL (s̄iPLbj)(s̄jPLbi). (34) The Wilson coefficients at the high scale are obtained by matching the ef- fective Hamiltonian to the ∆B = 2 squark-gluino box diagrams like those shown in Figure 8. The result is given by7: CVRR = α2s m2g̃ ∑ AB ΓR∗sAΓ R bAΓ R∗ sBΓ R bB × [ −11 9 B1(xA,xB) − 1 9 B2(xA,xB) ] (35) CSLL = α2s m2g̃ ∑ AB ΓR∗sAΓ L bAΓ R∗ sBΓ L bB × [ −17 18 B2(xA,xB) ] (36) CS×LL = α2s m2g̃ ∑ AB ΓR∗sAΓ L bAΓ R∗ sBΓ L bB × [ 1 6 B2(xA,xB) ] , (37) where the loop-functions are defined in Appendix A, the Γ’s are defined in Equation (12), and again xA = m̃ 2 A/m 2 g̃. The leading Standard Model contribution to Bs mixing is induced by a top quark box diagram which yields the following effective Hamiltonian [14] HSMeff = CSMLL (s̄γµPLb)(s̄γµPLb), (38) 7This result differs from [20], but agrees with subsequent analyses, e.g. [16]. 16 bR g ~ g ~ bR sR sR bR ~ bR ~ sR ~ sR ~ (δ23)RR d (δ23)RR d b b L L g ~ g ~ sR sR b b R R ~ ~ b b L L ~ ~ s s R R ~ ~ δ d 23( )RR tanβµmb tanβµmb δ d 23( )RR Figure 8: Diagrams contributing to Bs-mixing through large b̃R-s̃R mixing. The diagram on the right induces effective b̃L-s̃R mixing which contributes to CSLL and C S× LL . with the Wilson coefficient matched at MW CSMLL = G2F 16π2 M2W (V ∗ tsVtb) 2 S(m2t/M 2 W ), (39) where S(x) = 4x− 11x2 + x3 4(1 − x)2 − 2x3 lnx 2(1 − x)3 . (40) Before taking the hadronic matrix element of the effective Hamiltonian we must first take QCD corrections into account by using the renormalization group equations (RGE) to evolve the Wilson coefficients down to the low scale. The general NLO running of the Wilson coefficients for a ∆B = 2 effective Hamiltonian is given in [21] and [22] and involves mixing among different coefficients. In our case only the two scalar left-left operators mix, while the vector right-right and the vector-left-left coefficients simply scale multiplicatively. For simplicity we have evolved both operators from MW down to the mass of the b-quark.8 The hadronic matrix element of the effective Hamiltonian between Bs and B̄s states was calculated on the lattice [23] 〈B̄s |(s̄γµPRb)(s̄γµPRb)|Bs〉 = 1 3 mBsf 2 Bs B1(µ) (41) 8The supersymmetric contribution should in fact run from ∼ mg̃ down to mb. In this approximation we are ignoring corrections of order 1 − αs(mg̃)/αs(MW ) and potential contributions from the top quark in loops which is smaller. These corrections are part of the systematic uncertainty in our calculation. 17 〈B̄s |(s̄PLb)(s̄PLb)|Bs〉 = − 5 24 ( MBs mb + ms )2 mBsf 2 Bs B2(µ) (42) 〈B̄s |(s̄iPLbj)(s̄jPLbi)|Bs〉 = 1 24 ( MBs mb + ms )2 mBsf 2 Bs B3(µ), (43) with fBs as given in Table 1, B1(mb) = 0.87(4) +5 −4, B2(mb) = 0.82(3)(4), and B3(mb) = 1.02(6)(9), where the first error is statistical and the second is systematic, excluding uncertainty due to quenching. The quark masses in the above expression should be evaluated at the scale µ. The hadronic matrix element for the left-left current operator of the Standard Model is identical to that of the right-right operator shown in Equation (41). Finally, we can write the expression for the mass difference between Bs and B̄s as ∆MBs = 2 ∣∣∣〈B̄s ∣∣∣H∆B=2eff ∣∣∣Bs〉 ∣∣∣ . (44) The Standard Model and supersymmetric contributions interfere, H∆B=2eff = HSMeff + HSUSYeff . The input parameters used in the calculation are given in Table 1. Our results should be compared to the standard model prediction which can be obtained roughly by taking H∆B=2eff = HSMeff in Equation (44), which yields ∆MSMBs ∼ 13.8 ps−1. A more rigorous treatment given in [24] yields ∆MSMBs = 17.3 +1.5 −0.7 ps −1. (45) However, given the substantial uncertainty in the lattice evaluation of, e.g., fBs, it is probably appropriate to inflate this error, likely to the 25% level [25]. The current experimental limit, combining results from the LEP experiments and SLD, is [26] ∆MBs > 14.4 ps −1 (95 % confidence level). (46) Current and upcoming experiments are expected to be sensitive to mass dif- ferences much greater than the Standard Model prediction shown in Equa- tion (45). At Run II of the Tevatron [27] CDF is expected to probe up to ∆MBs of 41 ps −1 while BTeV is expected to achieve sensitivity to values up to ∆MBs ∼ 55 ps−1. Any evidence that ∆MBs > 25 ps−1 from these experiments would be a clear signal of new physics. To illustrate our results we add contours of constant ∆MBs (red dashed lines) to Figures 3-6. In the case of dominant right-right contributions, i.e. 18 Parameter Value Parameter Value Parameter Value mb 4.2 GeV mBs 5.379 GeV V ∗ tsVtb −0.04 mt 174 GeV fBs 204 MeV αs(MZ) 0.1185 MW 80.4 GeV MZ 91.2 GeV τBs 1.461 ps Table 1: Input parameters used in the calculation of SφK and ∆MBs. small µtanβ, the trend is similar to that of the previous section; lighter gluino and squarks give a larger SUSY contribution and thus increase ∆MBs . Note that the supersymmetric contribution to the mass difference dominates over the Standard Model in significant regions of the supersymmetric parameter space, easily allowing ∆MBs > 100 ps −1 where right-right mixing dominates. Such values for ∆MBs are certainly beyond the reach of the experiments mentioned above. In the case of dominant LR + RR mixing the modification of Bs mixing is not as striking. In the example given in Figure 6 values of ∆MBs are much closer to the standard model prediction. Restricting ourselves to areas that respect the b → sγ bound gives a yet lower value, within the reach of upcom- ing experiments. We should point out that this is not generic since Figure 6 only represents a slice of parameter space. Other choices of parameters can give higher values of ∆MBs (above 50 ps −1) for high values of µtanβ. The correlation of these results to those in B0d → φKS will be discussed in the next section. Finally, we should comment about possible CP violation in the Bs-B̄s system. In the standard CKM scenario the Bs → B̄s amplitude does not have a CP violating phase (in the Wolfenstein parameterization), so no indirect CP violation is expected. In our scenario, however, Bs mixing can involve the phases from the down-squark mass matrix. In the cases where the SUSY contribution to Bs mixing dominates the SM, measurements of CP violation in Bs → J/ψφ will be sensitive to these phases. 4 Correlation In this section we will discuss the correlation between SφK and ∆MBs in the context of large b̃-s̃ mixing. Because the effect on Bs mixing is very different for the two limiting regions of parameter space, we will discuss 19 them separately. 4.1 Dominant RR Mixing In this region of parameter space the operators O′3−6 make large contributions to SφK, while there is essentially only one contribution to Bs mixing, namely that from the operator shown in Equation (35). Unfortunately there is no simple, precise, relationship between the combination of the ∆B = 1 oper- ators and the operator responsible for Bs mixing. In general, they depend quite differently on loop functions. In spite of this, one can make the following strong statement. In cases where there is a large shift in SφK away from the Standard Model expectation due to the operators O′3−6, and the RR contribution to O′g (the dominant right-right mixing scenario), there is a large contribution to Bs mixing. To see this, we first note that the squarks and gluino must not be too heavy, and the b̃-s̃ mixing must be large in order to have a large contribution to SφK. This suggests a minimum contribution to the Bs mixing. However, there is the worry that it might be possible to fine-tune parameters to somehow drastically suppress the contribution to Bs mixing; for example, by choosing squark and gluino mass ratios to minimize the value of the functions B1 and B2 in Equation (35). We find that this is not possible, however. In order to have a very large contribution to SφK, one is pushed into a region of pa- rameter space where the gluino, and at least one of the down-type squarks is light. Furthermore, the splitting between this light squark, which represents a mixture of b̃ and s̃ squarks, and the masses of the heavier squarks must be large to avoid a super-GIM cancellation. Once this qualitative picture for the spectrum is identified, it is easy to check that there cannot be a cancellation of the contribution to Bs mixing in this case. Now we present our results quantitatively. Since our goal here will be to show that large deviations in SφK will correspond to large contributions to Bs mixing, we plot the minimum achievable value of ∆MBs for a given value of SφK. The minimum is found by scanning a parameter space that consists of the parameters {mg̃,m̃L2,m̃R2,m̃R3,cosθ56,φ6}. As discussed in Equation (12), cosθ56 represents the mixing angle between the right-handed b̃ and s̃ squarks, and φ6 represents the phase corresponding to this off-diagonal term. As a parameter space, we take: mg̃ ∈ (200,700) GeV (47) 20 m̃L2 ∈ (300,2500) GeV m̃R2 ∈ (300,2500) GeV m̃R3 ∈ (250,1000) GeV θ56 ∈ (0,π/2) φ6 ∈ (0,2π). The lower limits on the masses are motivated by direct searches, while the upper limits are motivated by naturalness considerations. A scan would generate a scatter plot of SφK vs. ∆MBs. For a given resultant value of SφK, we find the combination of parameters that yields the smallest contribution to Bs mixing. This is essentially equivalent to taking the boundary of the region generated by the scatter plot. As discussed in Section 2, there is considerable dependence on the variable κ, which has a relatively large uncertainty. So we repeat the above exercise for several values of κ, displaying the results in Figure 9. Adding the constraint from b → sγ modifies these results as shown in Figure 10. The contours in Figure 10 notably do not extend as low in SφK because the b → sγ constraint removes the region of parameter space that allowed us to obtain those values in Figure 9. The take-home message from the figures is a simple one. If the hint of the deviation in SφK measured in the B → φKS persists and it is attributable to a scenario with dominant RR squark mixing, it will result in a large contribution to Bs mixing, which will be a clear indication of new physics observable at the Tevatron. 4.2 Dominant LR + RR Mixing In the region of parameter space where µtanβ is relatively large, the expec- tation for ∆MBs is very different. In this region the main contribution to SφK comes from the LR + RR contribution to the dipole operator O′g. This contribution can be sizeable even when the squarks and gluinos are heavy (squarks can be at the TeV level or higher). This is the significant difference between the two limiting cases. Heavy squarks and gluino mean that the contributions from the operators O′3−6 are small. Similarly, the operators re- sponsible for Bs mixing, which come from box diagrams and resemble O′3−6, can also be small. The bottom line is that a large contribution to SφK is possible without a large addition to ∆MBs. This is borne out numerically, 21 0 100 200 300 400 500 600 700 800 −0.5 0 0.5 1−1 ∆ ∆ B B φ φ S Ss s κ= −2.0 κ= −1.5 κ= −1.1 M M −1 vs. γNo b s K K (ps ) Figure 9: The minimum value of ∆MBs for a given value of SφK. It is found by scanning over the parameter space given in Equation (47). The corresponding curve is shown for several values of κ, the coefficient of the dipole operator, as defined in Equation (25). 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1−0.2 ∆ ∆ B B φ φ S S s s κ= −1.1 κ= −1.5 κ= −2.0 M M −1 vs. K (ps ) Figure 10: The same as Figure 9, but with the b → sγ constraint applied as discussed in the text. The minimum value of ∆MBs for a given value of SφK. It is found by scanning over the parameter space given in Equation (47). The corresponding curve is shown for several values of κ, the coefficient of the dipole operator, as defined in Equation (25). 22 1 2 3 4 5 6 7 8 9 10 25 50 75 100 125 150 175 200 ∆MB s (ps )−1 tanµ β (TeV) 10 20 30 40 50 60 25 50 75 100 125 150 175 200 ∆MB s (ps )−1 tanµ β (TeV) Figure 11: Maximum and minimum values of ∆MBs as a function of µtanβ resulting from a scan of 55000 points in the parameter space with SφK < 0.3 and κ = −1.1. The plot on the left shows an enlargement of the region of small µtanβ. as shown in Figure 6, where the points allowed by the b → sγ constraint all give ∆MBs very close to the Standard Model expectation. We have seen that the LR + RR contribution to SφK can be important even for fairly small values of µtanβ, so it is natural to wonder what con- clusions can be drawn about ∆MBs in the regions where both contributions are important. To answer this question we again performed a scan of the pa- rameter space, this time collecting points with maximal and minimal ∆MBs as a function of µtanβ with the additional requirement that SφK < 0.3 for the nominal value κ = −1.1.9 The results are shown in Figure 11. In ac- cord with what was stated above, points with the largest µtanβ give smaller contributions to Bs mixing. In fact, for µtanβ > 30 TeV any effects on ∆MBs will be indistinguishable from the Standard Model expectation. This apparent upper bound may be interpreted as follows. For large µtanβ the severe b → sγ bound is pushing us to regions where RR mixing is small or masses are high, both of which disfavor large contributions to Bs mixing. At the other end of the spectrum, for µtanβ < 2 TeV all points in the scan gave values of ∆MBs > 30 ps −1, a clear signal of new physics above even the largest Standard Model predictions. This trend continuously connects us back to the result of the previous subsection. Mixing between the first two generations of squarks has fallen outside the main scope of this paper. We should mention, however, that larger values of the lightest squark masses, around 500 GeV, may well be preferred by 9The results are nearly identical when κ = −2.0 and SφK < −0.1. 23 constraints from K-K̄ mixing. To see this, note that if we believe that the Cabibbo angle originates through the down Yukawa matrices, then the down Yukawa matrix has the structure: λd = ( hd hsλC h hs ) , (48) where h is some unknown Yukawa coupling and λC is the sine of the Cabibbo angle. Diagonalizing this matrix requires a rotation on dL and sL of O(λC) which induces an off-diagonal element of order hdλC. Using the phenomeno- logical relationship, md/ms ≈ λ2C, we find that a rotation between dR and sR of O(λ3C) is needed to complete the diagonalization. Then, due to the lack of degeneracy between the squarks, the induced d̃-s̃ mixing can lead to a large contribution to the K-K̄ mixing. This suggests that heavier squark masses, perhaps above 500 GeV are preferred, barring some accidental cancellation with the (unknown) (1,2) element, h, in the above matrix. By the above reasoning, if one wishes to achieve an SφK that differs significantly from the value of sin 2β as measured in the in the B0d → J/ψKS channel, there may a theoretical prejudice to prefer the scenario where the LR+RR contributions dominate SφK, since in that case large squark masses are more easily accommodated while still giving a large effect due to the mg̃/mb enhancement. 5 Conclusion There exist a class of models, motivated by Grand Unified theories and the large observed mixing in atmospheric neutrinos, where it is natural to have a large mixing between the right-handed b̃ squark and s̃ squark. We have found that there exists a range of parameters where such mixing induces a significant deviation in SφK from the Standard Model expectation of sin 2β as measured in the channel B0d → J/ψKS. In particular, the central value for SφK from BaBar and BELLE can be accommodated without conflicting with the measured value of b → sγ, but there is some tension. The value of κ, the hadronic matrix element for the chromo-dipole operator, must be larger than the naive estimates for its value. There are two possible origins of a substantial modification to SφK. The first solely involves a large right-right mixing, with no contribution from the mixing proportional to mbµtanβ. In this case a small gluino mass is required, 24 near the experimental bound. Correspondingly, there is a large contribution to Bs mixing, a consequence which will be testable at the Tevatron Run II. In the second case, we consider the mixing from the combination of the large right-right mixing and a large mbµtanβ. In this case, squarks and gluinos need not be light, so Bs mixing need not be large. In particular, for very large values of µtanβ the prediction for Bs mixing is indistinguishable from the Standard Model prediction, when current errors on lattice matrix elements are taken into account. However, a substantial improvement in the Standard Model prediction for Bs mixing still may allow an effect to be seen at the Tevatron in this case. Note Added While completing this paper we received References [28, 29]. There is some overlap with these papers, which also consider supersymmetric contributions to B0d → φKS. Regarding [28], in places that we overlap, we agree qualitatively with their results, though there may be some quantitative differences. These are likely due to the fact that they work in the mass insertion approximation. Indeed, allowing large mixings and hierarchies that cannot be described by mass insertions gives us larger contributions in the ‘pure’ RR mixing case. Other possible differences may arise from a different treatment of the hadronic matrix elements and the fact that constraints from b → sγ were not imposed in the same way. We appear to disagree with [29], which seems to find a strong suppression of the contribution from the δRR23 mass insertion. We differ from both papers in our emphasis on the mixing induced by a combination of mbµtanβ along with a large flavor-changing RR element in the squark mass matrix. In those treatments, each mass insertion is consid- ered separately, including flavor off-diagonal LR mixing. We should point point out that the LR+RR mixing does not necessarily describe the same physics as a single LR flavor mixing insertion. Treating a LR + RR mixing as a pure δRL23 may miss important contributions due to the RR mixing only. For example, we find maximal values of ∆MBs for intermediate values of µtanβ which are much higher than those achieved in [28] with LR mixing. This might be due to a RR mixing contribution which is sub-dominant in SφK but which nevertheless gives a large contribution to 25 ∆MBs. This example illustrates how our framework differs from analyses which consider only one mass insertion at a time. Acknowledgments We would like to thank Zoltan Ligeti, Gustavo Burdman, and Yuval Gross- man for useful conversations. AP would also like to thank Gudrun Hiller for additional conversations. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contracts DE-AC03-76SF00098 and DE-AC03-76SF00515 and in part by the National Science Foundation under grant PHY-0098840. A Loop Functions We include the loop functions for completeness. We use the same definitions as in [11]. Here xA ≡ m2d̃A/m 2 g̃. B1(xA,xB) = ∫ ∞ 0 −1 4 z2 dz (z + 1)2(z + xA)(z + xB) (49) = − x 2 A logxA 4(xA −xB)(xA − 1)2 − x 2 B logxB 4(xB − xA)(xB − 1)2 − 1 4(xA − 1)(xB − 1) , (50) B2(xA,xB) = ∫ ∞ 0 zdz (z + 1)2(z + xA)(z + xB) (51) = − xA logxA (xA −xB)(xA − 1)2 − xB logxB (xB −xA)(xB − 1)2 − 1 (xA − 1)(xB − 1) , (52) C1(x) = 2x3 − 9x2 + 18x− 11 − 6 logx 36(1 − x)4 , (53) C2(x) = −16x3 + 45x2 − 36x + 7 + 6x2(2x− 3) logx 36(1 − x)4 . (54) 26 D1(x) = −x3 + 6x2 − 3x − 2 − 6x logx 6(1 −x)4 , (55) D2(x) = −x2 + 1 + 2x logx (x − 1)3 , (56) D3(x) = 2x3 + 3x2 − 6x + 1 − 6x2 logx 6(1 − x)4 , (57) D4(x) = −3x2 + 4x− 1 + 2x2 logx (x − 1)3 , (58) (59) B Chromo-dipole Matrix Element for B0d → φKS In this appendix we show explicitly the computation of the matrix element of O(′)g in the naive factorization approximation. The computation of a similar quantity for B̄0 → π+π− decay can be found in [30]. We start with Og = gs 8π2 mb(s̄iσ µνTaijPRbj)G a µν (60) and then connect a quark current through a virtual gluon to form a four- quark operator. This step depends on the convention used for the covariant derivative.10 Our convention is that Dµ = ∂µ +igT aAaµ, and we have checked that this is consistent with the Wilson coefficients for both the standard model and SUSY contributions. This yields the operator Og4 = i αs π kν k2 mb(s̄iσ µνTaijbj)(q̄kγµT a k�q�) (61) where k = pb − ps is the gluon momentum. In the naive factorization ap- proximation the color-octet current, (q̄kγµT a k�q�), cannot produce a physical φ, so the φ must be produced by the s̄ and q operators. To factor the matrix element we first use the equations of motion to simplify the tensor current. This yields: Og4 = − αs πk2 mb { mb(s̄γ µPLT a ijb) + ms(s̄γ µPRT a ijb) − 2pµb (s̄iPRTaijbj) } × (q̄kγµT a k�q�) (62) 10In particular, note that [31] appears to use the opposite convention of [30, 32]. 27 where the last term was simplified using the conservation of the quark cur- rent, kµ(q̄kγµT a k�q�) = 0 as in [30]. Then by a Fierz transformation and judi- cious use of Dirac matrix identities, this can be brought to the form [8, 31, 32] Og4 = − αs 2πk2 mb N2c − 1 2N2c ( δi�δkj − 2Nc N2c − 1 Tai�T a kj ) (63) × [2mb(s̄iγµPLq�)(q̄kγµPLbj) − 4mb(s̄iPRq�)(q̄kPLbj) +2ms(s̄iγ µPRq�)(q̄kγµPRbj) − 4ms(s̄iPLq�)(q̄kPRbj) +2(pb)µ {(s̄iγµPLq�)(q̄kPRbj) + (s̄iPRq�)(q̄kγµPRbj) +i(s̄iσ µνPRq�)(q̄kγ νPRbj) − i(s̄iγνPLq�)(q̄kσµνPRbj)}] Next we use the following parameterization for matrix elements. 〈φ(pφ, ,φ)|s̄γµs|0〉 = fφm2φ,µφ (64) 〈φ(pφ, ,φ)|s̄s|0〉 = 0 (65) 〈φ(pφ, ,φ)|s̄σµνs|0〉 = −ifφm2φ 2ms m2φ (, µ φp ν φ − ,νφpµφ). (66) Also, 〈K̄0(pK)|s̄γµb|B̄0d(pB)〉 = (pB + pK)µF+(t) + (pB −pK)µF−(t) (67) 〈K̄0(pK)|s̄b|B̄0d(pB)〉 = δ−1bs ∆BKF+(t) + δ−1bs (pB −pK)2F−(t) (68) 〈K̄0(pK)|s̄σµν(1 ±γ5)b|B̄0d(pB)〉 = −2is(pµBpνK − pνBpµK) ± sεµνλσ(pBλpKσ −pBσpKλ). (69) Here t = (pB − pk)2, ∆BK ≡ m2B − m2K, and δbs ≡ mb − ms. Notice we have corrected the sign in Equation (66) compared to the similar expression in [31]. Heavy quark effective theory gives the relation s = (F+ −F−)/4mb [33]. We also make the kinematic assumptions that the b-quark carries all of the B- meson momentum and that the φ momentum is equally divided between its two constituent s-quarks. Thus pb = pB and k 2 = 1 2 (m2B − m2φ/2 + m2K). Putting all the pieces together gives: κ = −m 2 b 2k2  1 + m 2 B − m2K + m2φ F−F+ 4mb(mb − ms)   + −m 2 b 2k2 [ m2K − m2B − 2m2φ 8m2b ( 1 − F− F+ ) + ms 2mb ( F− F+ )] (70) 28 Because the matrix element is nonsingular we have F−(0) = 0. Then for small t, due to simple pole dominance we have F−(m2φ) � F−(0) = 0. 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